76,739 research outputs found
On the minor-minimal 2-connected graphs having a fixed minor
Let H be a graph with κ1 components and κ 2 blocks, and let G be a minor-minimal 2-connected graph having H as a minor. This paper proves that |E(G)|-|E(H)|≤α(κ 1-1)+β(κ2-1) for all (α,β) such that α+β≥5,2α+5β≥;20, and β≥3. Moreover, if one of the last three inequalities fails, then there are graphs G and H for which the first inequality fails. © 2003 Elsevier B.V. All rights reserved
Large induced subgraphs via triangulations and CMSO
We obtain an algorithmic meta-theorem for the following optimization problem.
Let \phi\ be a Counting Monadic Second Order Logic (CMSO) formula and t be an
integer. For a given graph G, the task is to maximize |X| subject to the
following: there is a set of vertices F of G, containing X, such that the
subgraph G[F] induced by F is of treewidth at most t, and structure (G[F],X)
models \phi.
Some special cases of this optimization problem are the following generic
examples. Each of these cases contains various problems as a special subcase:
1) "Maximum induced subgraph with at most l copies of cycles of length 0
modulo m", where for fixed nonnegative integers m and l, the task is to find a
maximum induced subgraph of a given graph with at most l vertex-disjoint cycles
of length 0 modulo m.
2) "Minimum \Gamma-deletion", where for a fixed finite set of graphs \Gamma\
containing a planar graph, the task is to find a maximum induced subgraph of a
given graph containing no graph from \Gamma\ as a minor.
3) "Independent \Pi-packing", where for a fixed finite set of connected
graphs \Pi, the task is to find an induced subgraph G[F] of a given graph G
with the maximum number of connected components, such that each connected
component of G[F] is isomorphic to some graph from \Pi.
We give an algorithm solving the optimization problem on an n-vertex graph G
in time O(#pmc n^{t+4} f(t,\phi)), where #pmc is the number of all potential
maximal cliques in G and f is a function depending of t and \phi\ only. We also
show how a similar running time can be obtained for the weighted version of the
problem. Pipelined with known bounds on the number of potential maximal
cliques, we deduce that our optimization problem can be solved in time
O(1.7347^n) for arbitrary graphs, and in polynomial time for graph classes with
polynomial number of minimal separators
Complexity of Grundy coloring and its variants
The Grundy number of a graph is the maximum number of colors used by the
greedy coloring algorithm over all vertex orderings. In this paper, we study
the computational complexity of GRUNDY COLORING, the problem of determining
whether a given graph has Grundy number at least . We also study the
variants WEAK GRUNDY COLORING (where the coloring is not necessarily proper)
and CONNECTED GRUNDY COLORING (where at each step of the greedy coloring
algorithm, the subgraph induced by the colored vertices must be connected).
We show that GRUNDY COLORING can be solved in time and WEAK
GRUNDY COLORING in time on graphs of order . While GRUNDY
COLORING and WEAK GRUNDY COLORING are known to be solvable in time
for graphs of treewidth (where is the number of
colors), we prove that under the Exponential Time Hypothesis (ETH), they cannot
be solved in time . We also describe an
algorithm for WEAK GRUNDY COLORING, which is therefore
\fpt for the parameter . Moreover, under the ETH, we prove that such a
running time is essentially optimal (this lower bound also holds for GRUNDY
COLORING). Although we do not know whether GRUNDY COLORING is in \fpt, we
show that this is the case for graphs belonging to a number of standard graph
classes including chordal graphs, claw-free graphs, and graphs excluding a
fixed minor. We also describe a quasi-polynomial time algorithm for GRUNDY
COLORING and WEAK GRUNDY COLORING on apex-minor graphs. In stark contrast with
the two other problems, we show that CONNECTED GRUNDY COLORING is
\np-complete already for colors.Comment: 24 pages, 7 figures. This version contains some new results and
improvements. A short paper based on version v2 appeared in COCOON'1
Densities of Minor-Closed Graph Families
We define the limiting density of a minor-closed family of simple graphs F to
be the smallest number k such that every n-vertex graph in F has at most
kn(1+o(1)) edges, and we investigate the set of numbers that can be limiting
densities. This set of numbers is countable, well-ordered, and closed; its
order type is at least {\omega}^{\omega}. It is the closure of the set of
densities of density-minimal graphs, graphs for which no minor has a greater
ratio of edges to vertices. By analyzing density-minimal graphs of low
densities, we find all limiting densities up to the first two cluster points of
the set of limiting densities, 1 and 3/2. For multigraphs, the only possible
limiting densities are the integers and the superparticular ratios i/(i+1).Comment: 19 pages, 4 figure
Beyond Bidimensionality: Parameterized Subexponential Algorithms on Directed Graphs
We develop two different methods to achieve subexponential time parameterized
algorithms for problems on sparse directed graphs. We exemplify our approaches
with two well studied problems.
For the first problem, {\sc -Leaf Out-Branching}, which is to find an
oriented spanning tree with at least leaves, we obtain an algorithm solving
the problem in time on directed graphs
whose underlying undirected graph excludes some fixed graph as a minor. For
the special case when the input directed graph is planar, the running time can
be improved to . The second example is a
generalization of the {\sc Directed Hamiltonian Path} problem, namely {\sc
-Internal Out-Branching}, which is to find an oriented spanning tree with at
least internal vertices. We obtain an algorithm solving the problem in time
on directed graphs whose underlying
undirected graph excludes some fixed apex graph as a minor. Finally, we
observe that for any , the {\sc -Directed Path} problem is
solvable in time , where is some
function of \ve.
Our methods are based on non-trivial combinations of obstruction theorems for
undirected graphs, kernelization, problem specific combinatorial structures and
a layering technique similar to the one employed by Baker to obtain PTAS for
planar graphs
Complexity of the Steiner Network Problem with Respect to the Number of Terminals
In the Directed Steiner Network problem we are given an arc-weighted digraph
, a set of terminals , and an (unweighted) directed
request graph with . Our task is to output a subgraph of the minimum cost such that there is a directed path from to in
for all .
It is known that the problem can be solved in time
[Feldman&Ruhl, SIAM J. Comput. 2006] and cannot be solved in time
even if is planar, unless Exponential-Time Hypothesis
(ETH) fails [Chitnis et al., SODA 2014]. However, as this reduction (and other
reductions showing hardness of the problem) only shows that the problem cannot
be solved in time unless ETH fails, there is a significant
gap in the complexity with respect to in the exponent.
We show that Directed Steiner Network is solvable in time , where is a constant depending solely on the
genus of and is a computable function. We complement this result by
showing that there is no algorithm for
any function for the problem on general graphs, unless ETH fails
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